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Linear recurrence relations for cluster variables of affine quivers. (English) Zbl 1252.16012

From the introduction: Our main motivation in this paper comes from a conjecture formulated by I. Assem, C. Reutenauer and D. Smith [Adv. Math. 225, No. 6, 3134-3165 (2010; Zbl 1275.13017)]: They proved that if the frieze sequences associated with a (valued) quiver \(Q\) satisfy linear recurrence relations, then \(Q\) is necessarily affine or Dynkin. They conjectured that conversely, the frieze sequences associated with a quiver of Dynkin or affine type always satisfy linear recurrence relations. For Dynkin quivers, this is immediate from Fomin-Zelevinsky’s classification theorem for the finite-type cluster algebras [S. Fomin, A. Zelevinsky, Invent. Math. 154, No. 1, 63-121 (2003; Zbl 1054.17024)]. Assem, Reutenauer and Smith [loc. cit.] gave an ingenious proof for the affine types \(\widetilde A\) and \(\widetilde D\) as well as for the non-simply laced types obtained from these by folding. For the exceptional affine types, the conjecture remained open.
In this paper, we prove Assem-Reutenauer-Smith’s conjecture in full generality using the representation-theoretic approach to cluster algebras pioneered by R. Marsh, M. Reineke, and A. Zelevinsky [Trans. Am. Math. Soc. 355, No. 10, 4171-4186 (2003; Zbl 1042.52007)]. More precisely, our main tool is the categorification of acyclic cluster algebras via cluster categories [cf., e.g., B. Keller, Lond. Math. Soc. Lect. Note Ser. 375, 76-160 (2010; Zbl 1215.16012)] and especially the cluster multiplication formula of P. Caldero and B. Keller [Ann. Sci. Éc. Norm. Supér. (4) 39, No. 6, 983-1009 (2006; Zbl 1115.18301)]. Our method also yields a new proof for \(\widetilde A\) and \(\widetilde D\). It leads to linear recurrence relations which are explicit for the frieze sequences associated with the extending vertices and which allow us to conjecture explicit minimal linear recurrence relations for all vertices.

MSC:

16G20 Representations of quivers and partially ordered sets
13F60 Cluster algebras
18E30 Derived categories, triangulated categories (MSC2010)

Software:

quivermutation
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Full Text: DOI arXiv

References:

[1] Assem, Ibrahim; Dupont, Grégoire, Friezes and a construction of the Euclidean cluster variables, J. Pure Appl. Algebra, 215, 10, 2322-2340 (2011) · Zbl 1317.13049
[2] Assem, Ibrahim; Reutenauer, Christophe; Smith, David, Friezes, Adv. Math., 225, 6, 3134-3165 (2010) · Zbl 1275.13017
[3] Assem, Ibrahim; Simson, Daniel; Skowroński, Andrzej, Elements of the Representation Theory of Associative Algebras, vol. 1: Techniques of Representation Theory, London Math. Soc. Stud. Texts, vol. 65 (2006), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1092.16001
[4] Bakke Buan, Aslak; Marsh, Robert J.; Reineke, Markus; Reiten, Idun; Todorov, Gordana, Tilting theory and cluster combinatorics, Adv. Math., 204, 2, 572-618 (2006) · Zbl 1127.16011
[5] Berstel, Jean; Reutenauer, Christophe, Noncommutative Rational Series with Applications, Encyclopedia Math. Appl., vol. 137 (2011), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1250.68007
[6] Caldero, Philippe; Chapoton, Frédéric, Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv., 81, 3, 595-616 (2006) · Zbl 1119.16013
[7] Caldero, Philippe; Keller, Bernhard, From triangulated categories to cluster algebras. II, Ann. Sci. Ecole Norm. Sup. (4), 39, 6, 983-1009 (2006) · Zbl 1115.18301
[8] Cecotti, Sergio; Vafa, Cumrun, Classification of complete \(N = 2\) supersymmetric field theories in 4 dimensions · Zbl 1320.81085
[9] Conway, J. H.; Coxeter, H. S.M., Triangulated polygons and frieze patterns, Math. Gaz., 57, 400, 87-94 (1973) · Zbl 0285.05028
[10] Conway, J. H.; Coxeter, H. S.M., Triangulated polygons and frieze patterns, Math. Gaz., 57, 401, 175-183 (1973) · Zbl 0288.05021
[11] Coxeter, H. S.M., Frieze patterns, Acta Arith., 18, 297-310 (1971) · Zbl 0217.18101
[12] William Crawley-Boevey, Lectures on representations of quivers, unpublished notes.; William Crawley-Boevey, Lectures on representations of quivers, unpublished notes. · Zbl 0857.16014
[13] Di Francesco, Philippe; Kedem, Rinat, \(Q\)-system cluster algebras, paths and total positivity, SIGMA Symmetry Integrability Geom. Methods Appl., 6 (2010), Paper 014, 36 p · Zbl 1241.13020
[14] Di Francesco, Philippe; Kedem, Rinat, \(Q\)-systems, heaps, paths and cluster positivity, Comm. Math. Phys., 293, 3, 727-802 (2010) · Zbl 1194.05165
[15] Dlab, Vlastimil; Ringel, Claus Michael, Representations of Graphs and Algebras, Carleton Math. Lect. Notes, vol. 8 (1974), Department of Mathematics, Carleton University: Department of Mathematics, Carleton University Ottawa, ON · Zbl 0449.16022
[16] Dupont, G., Generic variables in acyclic cluster algebras, J. Pure Appl. Algebra, 215, 4, 628-641 (2011) · Zbl 1209.13024
[17] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras. I. Foundations, J. Amer. Math. Soc., 15, 2, 497-529 (2002), (electronic) · Zbl 1021.16017
[18] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras. II. Finite type classification, Invent. Math., 154, 1, 63-121 (2003) · Zbl 1054.17024
[19] Fomin, Sergey; Zelevinsky, Andrei, \(Y\)-systems and generalized associahedra, Ann. of Math. (2), 158, 3, 977-1018 (2003) · Zbl 1057.52003
[20] Fordy, Allan; Marsh, Robert, Cluster mutation-periodic quivers and associated Laurent sequences, J. Algebraic Combin., 1-48 (2010)
[21] Happel, Dieter, Auslander-Reiten triangles in derived categories of finite-dimensional algebras, Proc. Amer. Math. Soc., 112, 3, 641-648 (1991) · Zbl 0736.16005
[22] Bernhard Keller, Quiver mutation in Java, Java applet available at the authorʼs home page.; Bernhard Keller, Quiver mutation in Java, Java applet available at the authorʼs home page.
[23] Keller, Bernhard, On triangulated orbit categories, Doc. Math., 10, 551-581 (2005) · Zbl 1086.18006
[24] Keller, Bernhard, Cluster algebras, quiver representations and triangulated categories, (Holm, Thorsten; Jørgensen, Peter; Rouquier, Raphaël, Triangulated Categories. Triangulated Categories, London Math. Soc. Lecture Note Ser., vol. 375 (2010), Cambridge University Press), 76-160 · Zbl 1215.16012
[25] Lampe, Philipp, A quantum cluster algebra of Kronecker type and the dual canonical basis, Int. Math. Res. Notices, 2011, 13, 2970-3005 (2011) · Zbl 1273.17020
[26] Marsh, Robert; Reineke, Markus; Zelevinsky, Andrei, Generalized associahedra via quiver representations, Trans. Amer. Math. Soc., 355, 10, 4171-4186 (2003), (electronic) · Zbl 1042.52007
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